How Synchronization Protects from Noise
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Tabareau, Nicolas, Jean-Jacques Slotine, and Quang-Cuong
Pham. “How Synchronization Protects from Noise.” PLoS
Comput Biol 6.1 (2010): e1000637. © 2010 Tabareau et al.
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http://dx.doi.org/10.1371/journal.pcbi.1000637
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How Synchronization Protects from Noise
Nicolas Tabareau1,2*, Jean-Jacques Slotine3, Quang-Cuong Pham1
1 LPPA, Collège de France, Paris, France, 2 INRIA, École des mines de Nantes, France, 3 Nonlinear Systems Laboratory, Massachusetts Institute of Technology, Cambridge,
Massachusetts, United States of America
Abstract
The functional role of synchronization has attracted much interest and debate: in particular, synchronization may allow
distant sites in the brain to communicate and cooperate with each other, and therefore may play a role in temporal binding,
in attention or in sensory-motor integration mechanisms. In this article, we study another role for synchronization: the socalled ‘‘collective enhancement of precision’’. We argue, in a full nonlinear dynamical context, that synchronization may help
protect interconnected neurons from the influence of random perturbations—intrinsic neuronal noise—which affect all
neurons in the nervous system. More precisely, our main contribution is a mathematical proof that, under specific,
quantified conditions, the impact of noise on individual interconnected systems and on their spatial mean can essentially be
cancelled through synchronization. This property then allows reliable computations to be carried out even in the presence
of significant noise (as experimentally found e.g., in retinal ganglion cells in primates). This in turn is key to obtaining
meaningful downstream signals, whether in terms of precisely-timed interaction (temporal coding), population coding, or
frequency coding. Similar concepts may be applicable to questions of noise and variability in systems biology.
Citation: Tabareau N, Slotine J-J, Pham Q-C (2010) How Synchronization Protects from Noise. PLoS Comput Biol 6(1): e1000637. doi:10.1371/journal.pcbi.1000637
Editor: Michael Breakspear, The University of New South Wales, Australia
Received June 18, 2009; Accepted December 8, 2009; Published January 15, 2010
Copyright: ß 2010 Tabareau et al. This is an open-access article distributed under the terms of the Creative Commons Attribution License, which permits
unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited.
Funding: This work was supported in part by EC—contract number FP6-IST-027140, action line: Cognitive Systems. The funders had no role in study design, data
collection and analysis, decision to publish, or preparation of the manuscript.
Competing Interests: The authors have declared that no competing interests exist.
* E-mail: nicolas.tabareau@inria.fr
The influence of noise on the behaviors of nonlinear systems
is very diverse. In chaotic systems, a small amount of noise can
yield dramatic effects. At the other end of the spectrum, the
effect of noise on nonlinear contracting systems is bounded by
s2 =l where s is the noise intensity – which can be arbitrarily
large – and l is the contraction rate of the system [21]. Between
these two extremes, it has been shown analytically that some
limit-cycle oscillators commonly used as simplified neuron
models, such as FitzHugh-Nagumo (FN) oscillators, are basically
unperturbed when they are subject to a small amount of white
noise [22]. Yet, a larger amount of noise breaks this
‘‘resistance’’, both in the state space and in the frequency space
[Figures 1(A)–(D)]. This suggests that both temporal coding and
frequency coding may be unusable in the context of large
neuronal noise.
One might argue that it could be possible to recover some
information from the noisy FN oscillators by considering the
activities of a large number of oscillators simultaneously [19,23].
Figure 2(A) shows that the spatial mean of the noisy oscillators still
carries very little information when the noise intensities are large,
making the population coding hypothesis also unlikely in this
context. In other words, if the underlying dynamics are
fundamentally nonlinear, as in the case of our FN oscillators, the
spatial mean of the signals is ‘‘clean,’’ but contains very little
information: the nonlinear nature of the systems dynamics
prevents the familiar ‘‘averaging out’’ of noise through multiple
measurements, and getting rid of the noise also gets rid of the
signal.
By contrast, one can observe that when oscillators are
synchronized through mutual couplings, then they become ‘‘protected’’ from noise, whether in temporal [Figure 1(E)], frequential
Introduction
Synchronization phenomena are pervasive in biology. In
neuronal networks [1–3], a large number of studies have sought
to unveil the mechanisms of synchronization, from both
physiological [4,5] and computational viewpoints (see for instance
[6] and references therein). In addition, the functional role of
synchronization has also attracted considerable interest and
debates. In particular, synchronization may allow distant sites in
the brain to communicate and cooperate with each other [7–9]
and therefore may play a role in temporal binding [10,11] and in
attention and sensory-motor integration mechanisms [12–14].
In this article, we study another role for synchronization: the socalled collective enhancement of precision (see e.g. [15–17]), an intuitive
and often quoted phenomenon with comparatively little formal
analysis [18]. We explain mathematically why synchronization
may help protect interconnected nonlinear dynamic systems from
the influence of random perturbations. In the case of neurons,
these perturbations would correspond to so-called ‘‘intrinsic
neuronal noise’’ [19], which affect all of the neurons in the
nervous system. In the presence of significant noise intensities (as
experimentally found in e.g. retinal ganglion cells in primates
[20]), this property would be required for meaningful and reliable
computations to be carried out.
It should be noted that ‘‘protection of systems from noise’’ and
‘‘robustness of synchronization to noise’’ are two different
concepts. The latter concept means that the synchronized systems
remain so in presence of noise, whereas the former concept means
that, thanks to synchronization, the behaviors of the coupled
systems are close to the noise-free behaviors. This difference is
further addressed in the Discussion.
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How Synchronization Protects from Noise
sum of the outgoing connection weights
Author Summary
Synchronization phenomena are pervasive in biology,
creating collective behavior out of local interactions
between neurons, cells, or animals. On the other hand,
many of these systems function in the presence of large
amounts of noise or disturbances, making one wonder
how meaningful behavior can arise in these highly
perturbed conditions. In this paper we show mathematically, in a general context, that synchronization is actually
a means to protect interconnected systems from effects of
noise and disturbances. One possible mechanism for
synchronization is that the systems jointly create and then
share a common signal, such as a mean electrical field or a
global chemical concentration, which in turn makes each
system directly connected to all others. Conversely,
extracting meaningful information from average measurements over populations of cells (as commonly used for
instance in electro-encephalography, or more recently in
brain-machine interfaces) may require the presence of
synchronization mechanisms similar to those we describe.
Vi
j
Kij :
j
Quorum sensing, and more generally the measurement of a
common mean signal, can thus be seen as a practical (and
biologically plausible) way to implement all-to-all coupling with 2n
connections instead of n2 .
(A2). Let Hj denote the Hessian matrix of the function fj and
let lmax (Hj ) denote its largest eigenvalue. For all j, we assume that
1
lmax (Hj ) is uniformly upper-bounded by a constant pffiffiffi Hbd . This
d
implies in particular that
Vx,j,t
Hbd
xT Hj xƒ pffiffiffi ExE2 :
d
This assumption gives us a bound on the nonlinearity of f, the
extreme case being Hbd ~0 for a linear system.
(A3). The dynamics f is resistant to small perturbations. More
precisely, consider two systems starting from the same initial
conditions but driven by slightly different dynamics
General analytical result
Consider a diffusive network of d-dimensional noisy non-linear
dynamical systems
x_ noise{free ~f(xnoise { free ,t)
!
Kji (xj {xi ) dtzsdWi , i~1 . . . n
X
dxi ~ðf(xi ,t)zk(x. {xi )ÞdtzsdWi :
Results
dxi ~ f(xi ,t)z
Kji ~
Remark that any symmetric network is balanced.
A particular kind of balanced network consists of an all-to-all
network with identical couplings, i.e. Kij ~k=n for all i and j. In
general, assuming all-to-all coupling needs not be unduly
restrictive, since such coupling can be implemented through
mechanisms such as quorum sensing [25–27]. Indeed, assuming that
the mean value of the xi ’s can be provided by the environment as
1X
x , then the all-to-all network (1) can be written as a
x. ~
i i
n
star network where damping is added locally and each cell xi is
only connected to the common signal
[Figure 1(F)] or ‘‘populational’’ aspects [Figure 2(B)]. Thus, in
some sense, the linear effect of averaging noise while preserving
signal [24] can be achieved for these highly nonlinear dynamic
components through the process of synchronization. Our aim in this
article is to give mathematical elements of explanation for this
phenomenon, in a full nonlinear setting. It is also to suggest
elements of response to a more general question, namely: what is
the precise meaning of ensemble measurements or population codes,
and what information do they convey about the underlying
dynamics and signals?
X
X
and
ð1Þ
j=i
x_ perturbed ~f(xperturbed ,t)zP,
where f~(f1 , . . . ,fd )T is a Rd ?Rd function. Note that the noise
intensity s is intrinsic to the dynamical system (i.e. independent of
the inputs), which is consistent with experimental findings [20].
For simplicity, we set s to be a constant in this article, although the
case of time- and state-dependent noise intensities can be easily
adapted from [21].
We consider four mathematical assumptions that will enable us
to relate the trajectory of any noisy element of the network xi to
the trajectory of the noise-free system xnoise { free driven by
equation
(where P is a real time stochastic process) then (EPE)?0 implies
Exnoise { free {xperturbed E?0.
In particular, such a property has been demonstrated in the case
of FN oscillators, with P representing a white noise process [22].
(A4). After exponential transients, the expected sum of the
squared distances between the states of the elements of the
network is bounded by a constant r
X
dxnoise { free ~f(xnoise { free ,t)dt:
ivj
This is where synchronization will come into play, because
synchronization is an effective way to reduce the bound r. Some
precise conditions for this will be given later.
We show in Methods that under these assumptions and when
n?? and r=n2 ?0, the distance between the trajectory of any
noisy element xi of the network and that of the noise-free system
xnoise { free tends to zero, with the impact of noise on the mean
(A1) is an assumption on the form of the network. (A2) gives a
bound on the nonlinearity of the dynamics f. (A3) states that the
system trajectories are resistant to small perturbations. Finally,
(A4) requires that the dynamical systems in the network are
synchronized.
(A1). The network is balanced, that is, for any element of the
network, the sum of the incoming connection weights equals the
PLoS Computational Biology | www.ploscompbiol.org
!
Exi {xj E2 ƒr:
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January 2010 | Volume 6 | Issue 1 | e1000637
How Synchronization Protects from Noise
Figure 1. Simulations of a network of FN oscillators using the Euler-Maruyama algorithm [47]. The dynamics of coupled FN oscillators are
given by equation (2). The parameters used in all simulations are a~0:3, b~0:2, c~30. (A) shows the trajectory of the ‘‘membrane potential’’ of a
noise-free oscillator and (B) depicts the frequency spectrum of this trajectory computed by Fast Fourier Transformation. (C) and (D) present the
trajectory (respectively the frequency spectrum) of a noisy uncoupled oscillator (s~10). (E) and (F) show the trajectory (respectively the frequency
spectrum) of a noisy synchronized oscillator within an all-to-all network (s~10, kij ~5, n~200). Note the temporal and frequential similarities between
a noise-free oscillator and a noisy synchronized one. For instance, the main frequency and the first harmonics are very similar in the two frequency
spectra. In contrast, the frequency spectrum of a noisy uncoupled oscillator shows no clear harmonics.
doi:10.1371/journal.pcbi.1000637.g001
trajectory evolving as
f(x,t)~A(t)xzb(t), we recover the known result [28] that the
impact of noise evolves as the inverse square root of n. More
generally, linear components of the system dynamics (including, in
particular, the input signals) do not contribute to the first term of
the above upper bound.
rHbd
s
z pffiffiffi :
2n2
n
In particular, when f is a time-varying linear system of the form
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How Synchronization Protects from Noise
Figure 2. ‘‘Spatial mean’’ of FN oscillators. Note that the same set of random initial conditions was used in the two plots. (A) shows the average
‘‘membrane potential’’ computed over n~200 noisy uncoupled oscillators (s~10). (B) shows the average ‘‘membrane potential’’ computed over n~200
noisy synchronized oscillators within an all-to-all network (s~10, kij ~5). Observe that, in the first plot, the average trajectory of uncoupled oscillators
carries essentially no information, while in the second plot, the average trajectory of synchronized oscillators is very similar to a noise-free one.
doi:10.1371/journal.pcbi.1000637.g002
This statement can be further tested by constructing a modelbased nonlinear state estimator (observer) [29]. Let (vi , wi )T be a
noisy synchronized oscillator and consider the observer
Synchronization in networks of noisy FN oscillators
We now give conditions to guarantee assumption (A4) for all-toall networks of FN oscillators with identical couplings. The
dynamics of n noisy FN oscillators coupled by (gap-junction-like)
diffusive connections is given by
8
>
>
>
< dvi
>
>
>
: dwi
~
~
!
Pk
(vj {vi ) dtzsdWi
cf (vi , wi , I)z
j n
8
< vobs
: wobs
If vi has the same trajectory as a noise-free FN oscillator, then it
can be shown that (vobs , wobs )T tends exponentially to (vi , wi )T ,
independently of the observer’s initial conditions [29]. Thus the
squared distance (vobs {vi )2 indicates how close vi is from a noisefree oscillator [see Figure 3(B) for a comparison this theoretical
result with simulations].
1
where f (v, w, I)~v{ v3 zwzI. We show in Methods that,
3
after exponential transients of rate k,
!
2
(vi {vj )
ivj
ƒ
n(n{1)s2
:
k
Simulations of more generic networks
ð3Þ
We provide in this section simulation results which show that
similar observations can be made even for more general network
classes that are not yet covered by the theory. We believe that this
simulations show the genericity of the concepts presented above.
Probabilistic networks. In practice, all-to-all neuronal
networks of large size are rare. Rather, the mechanisms of
neuronal connections in the brain are believed to be probabilisitic
in nature (see [30] for a review). Here, we consider a probabilistic
symmetric network of n oscillators, where any pair of oscillators
has probability p to be symmetrically connected and probability
1{p to be unconnected. Figure 4 shows simulation results for
randomly chosen network with p~0:1. Concretely, we have built
a network by randomly deciding for any pair of connections if the
connection exists or not.
Hindmarsh-Rose oscillators. Hindmarsh-Rose oscillators
are three-dimensional dynamical systems that are also often used
as neuron models
Thus, (A4) is verified with
r~
n(n{1)s2
:
k
ð4Þ
For large n, we have r=n2 *s2 =k, which converges to 0 when
k??. Figure 3(A) provides a comparison of this theoretical
bound with simulations.
Assumption (A1) is also verified because an all-to-all network
with identical couplings is symmetric, therefore balanced. Since
the (vi , wi )T are oscillators with stable limit cycles, it can be shown
that the trajectories of the vi are bounded by a common constant
M. Thus (A2) is verified with Hbd ~2cM. Finally, (A3) may be
adapted from [22]. Indeed, we believe that the arguments of [22]
can be extended to the case of non-white noise. Making this point
precise is the subject of ongoing work.
Using now the ‘‘general analytical result’’, we obtain that, given any
(non necessarily small) noise intensity s, in the limits for k?? and
n?? and after exponential transients, the behavior of any oscillator
will be arbitrary close to that of a noise-free oscillator (Figure 1).
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ð5Þ
ð2Þ
1
{ (vi {azbwi )dt
c
X
~ cf (vobs , wobs , I)zkobs (vi {vobs )
1
~ { (vobs {azbwobs ):
c
8
>
< dV
dn
>
:
dm
4
~ (I{n{m{V 3 zgV V zEV V 2 )dtzsdW
~ (GNa zENa V 2 {n)dt
~ (gCa (ECa (V zVconst ){m))dt
January 2010 | Volume 6 | Issue 1 | e1000637
How Synchronization Protects from Noise
Figure 3. Asymptotic appraisal of the theoretical bounds. Note that the experimental expectations were computed assuming the ergodic
1 X
(v {v. )2 ) as a function of the coupling
hypothesis. (A) Expectation of the average squared distance between the vi ’s and v. (given by
i i
n
2
(n{1)s
strength kij (s~10). Theoretical bound
(cf equations (7) and (4)) for n~10 (bold line), for n~50 (plain line), for n~200 (dashed line);
n2 kij
simulation results for n~10 (squares), for n~50 (triangles), for n~200 (crosses). (B) Expected squared distance between a noisy synchronized
(n{1)s2
was plotted in plain line and the simulation
oscillator and its observer (given by (vobs {vi )2 ) as a function of n (s~10, kij ~5). The bound
n2 kij
results were represented by crosses.
doi:10.1371/journal.pcbi.1000637.g003
Discussion
with gV ~0:5; EV ~2:8; GNa ~0; ENa ~4:4; gCa ~0:001; ECa ~9;
Vconst ~7=ECa. These oscillators can exhibit more complex
behaviors (including spiking and bursting regimes [31]) than
FitzHugh-Nagumo oscillators. The proofs of (A3) and (A4) for
Hindmarsh-Rose oscillators are the object of ongoing research.
We made the inputs time-varying in this simulation. In fact, all
the previous calculations can be straightforwardly extended to the
case of time-varying inputs, as long as those inputs are the same for
all the oscillators [6].
One can observe from the simulations (see Figure 5) that the
synchronized oscillators preserve the input signal, while the
uncoupled oscillators completely blur it out.
We have argued that synchronization may represent a
fundamental mechanism to protect neuronal assemblies from
noise, and have quantified this hypothesis using a simple nonlinear
neuron model. This may further strengthen our understanding of
synchronization in the brain as playing a key functional role,
rather than as being mostly an epiphenomenon.
It should be noted that the causal relationship studied here –
effect of synchronization on noise – is converse to one usually
investigated formally in the literature – effect of noise on
synchronization: under certain conditions, adding noise can desynchronize already synchronized oscillators (destructive effect)
Figure 4. Simulation for a probabilistic symmetric network (n~200, p~0:1, s~10, kij ~5). (A) shows the trajectory of the ‘‘membrane
potential’’ of an oscillator in the network. (B) shows its frequency spectrum. Compare these two plots with those in Figure 1.
doi:10.1371/journal.pcbi.1000637.g004
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How Synchronization Protects from Noise
Figure 5. Simulation of Hindmarsh-Rose oscillators with time varying inputs. (A) The time-varying input voltage. (B) Trajectory of the
‘‘membrane potential’’ of a noise-free oscillator. (C) Trajectory of a noisy uncoupled oscillator. (D) Trajectory of a noisy synchronized oscillator (n~200,
s~10, kij ~5).
doi:10.1371/journal.pcbi.1000637.g005
[32]; under other conditions, adding noise can, on the contrary,
synchronize oscillators that were not synchronized (constructive
effect) [33,34]; for a review, see [35]. Also, previous papers have
studied a similar phenomenon of improvement
in precision by
pffiffiffi
synchronization. Enright [28] shows n improvement in a model
of coupled relaxation oscillators, all interacting through a common
pffiffiffi
accumulator variable (possibly being the pineal gland). This n
improvement has been experimentally shown in real heart cells
pffiffiffi
[36]. More recently, [37] shows a way to get better than n
improvement. However, their studies primarily focused on the
case of phase oscillators, which are linear dynamical systems. In
contrast, we concentrate here on the more general case of
nonlinear oscillators, and quantify in particular the effect of the
oscillators’ nonlinearities. The assumptions we consider are also
different: while most existing approaches (including [37]) assume
weak couplings and small noise intensities, we consider here strong
couplings and arbitrary noise intensities.
The mechanisms highlighted in the paper may also underly
other types of ‘‘redundant’’ calculations in the presence of noise
and variability. In otoliths for instance, ten of thousands of hair
cells jointly compute the three components of acceleration [38,39].
In muscles, thousands of individual fibers participate in the control
of one single degree of freedom. Similar questions may also arise in
systems biology, e.g., in cell mechanisms of quorum sensing where
individual cells measure global chemical concentrations in their
environment in a fashion functionally similar to all-to-all coupling
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[25–27], in mechanical coupling of motor proteins [40], in the
context of transcription-regulation networks [41,42], and in
differentiation dynamics [43].
Finally, the results point to the general question: what is the
precise meaning of ensemble measurements or population codes,
what information do they convey about the underlying signals, and is
the presence of synchronization mechanisms (gap-junction mediated
or other) implicit in this interpretation? As such, they may also shed
light on a somewhat ‘‘dual’’ and highly controversial current issue.
Ensemble measurements from the brain can correlate to behavior,
and they have been suggested e.g. as inputs to brain-machine
interfaces. Are these ensemble signals actually available to the brain
[44], perhaps through some process akin to quorum sensing, and
therefore functionally similar to (local) all-to-all coupling? Are local
field potentials [45] plausible candidates for a role in this picture?
Methods
Proof of the general analytical result
In the noise-free case (s~0), it can be shown that, for strong
enough coupling strengths, the elements of the network synchronize completely, that is, after exponential transients, we have r~0
in (A4) [6]. Thus, all the xi tend to a common trajectory, which is
in fact a nominal trajectory of the noise-free system
x_ noise{free ~f(xnoise { free ,t), because all the couplings vanish on
the synchronization subspace.
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January 2010 | Volume 6 | Issue 1 | e1000637
How Synchronization Protects from Noise
1X
Turning now to the noise term
sdWi in Equation (10), we
i
n
have
In the presence of noise, it is not clear how to relate the
trajectory of each xi to a nominal trajectory of the noise-free
system. Nevertheless, we still know that the xi live ‘‘in a small
neighborhood’’ of each other, as quantified by (A4). Thus, if the
center of this small neighborhood follows a trajectory similar to a
nominal trajectory of the noise-free system, then one may gain
some information on the trajectories of the xi .
To be more precise, let x. be the center of mass of the xi , that is
1X
xi :
ð6Þ
x. ~
n i
1X
s
sdWi % pffiffiffi dW
n i
n
since the intrinsic noises of the elements of the network are
mutually independent.
Thus, for a given (even large) noise intensity s, the difference
between the dynamics followed by x. and the noise-free dynamics f
tends to zero when n?? and r=n2 ?0. Assumption (A3) then
implies that Ex. {xnoise { free E?0. More precisely, the impact of
noise on the mean trajectory (quantified by (Ex. {xnoise { free E))
evolves as
Observe that, after expansion and rearrangement, the sum
P
2
ivj Exi {xj E can be rewritten in terms of the distances of the
xi from x.
X
Exi {xj E2 ~n
X
ivj
rHbd
s
z pffiffiffi :
2n2
n
Exi {x. E2 :
i
X
Exnoise { free {xi EƒExnoise { free {x. EzEx. {xi E
!
r
ƒ :
n
. 2
Exi {x E
i
ð7Þ
FN oscillators in an all-to-all network
Two FN oscillators. Consider first the case of two coupled
FN oscillators driven by Equation (2). Construct the following
auxiliary system (or virtual system, in the sense of [46]), where v1
and v2 are considered as external inputs
!
1 X
1X
f(xi ,t) dtz
sdWi :
dx ~
n
n i
i
.
ð8Þ
We now make the dynamics explicit with respect to x. by letting
8
>
< dx1
!
n
1 X
f(xi ,t) {f(x. ,t)
n i~1
>
: dx2
ð9Þ
1X
sdWi :
n i
ð10Þ
Using the Taylor formula with integral remainder, we have
~
0
fj (xi ,t){fj (x. ,t){Fj (x. ,t)T (xi {x. )
(1{s)(xi {x. )T Hj ((1{s)xi zsx. )(xi {x. )ds
ð11Þ
where Fj is the gradient of fj or, equivalently, the j th vector of the
Jacobian matrix of f. Summing Equation (11) over i and using
assumption (A2), we get
j
X
i
Hbd X
(fj (xi ,t){fj (x. ,t))jƒ pffiffiffi
Exi {x. E2 :
2 d i
which implies that
rHbd
2n2
ð12Þ
ð18Þ
leading to
(
ð13Þ
pffiffiffi 00
0
dy1 ~{l1 y1 dtz 2sl1 dW
pffiffiffi
0
dy2 ~{l2 y2 dtz 2scdW :
Since these equations are in fact uncoupled, they can be
solved independently. Using the stochastic contraction results
(Ee E)?0 when r=n2 ?0.
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8
< y ~l00 x z 1 x
1
2
1 1
c
000
:
y2 ~cx1 zl1 x2
Summing now inequality (12) over j and using inequality (7), we
get
(Ee E)ƒ
pffiffiffi
(c{(v21 zv1 v2 zv22 ){k)x1 zcx2 ) dtz 2sdW
ð17Þ
1
b
~
{ x1 { x2 dt:
c
c
~
Remark that (x1 , x2 )T ~(v1 {v2 , w1 {w2 )T is a particular
trajectory of this system.
Let l1 ~kz(v21 zv1 v2 zv22 ){c and l2 ~b=c. Assume that the
coupling strength is significantly larger than the system’s
parameters, i.e. k&c, k&1=c and k&b=c. Since v21 zv1 v2 zv22
is nonnegative for any v1 and v2 , we have either l1 §k or l1 ^k,
depending on the actual value of v21 zv1 v2 zv22 . This implies in
particular that l1 &c, l1 &1=c and l1 &l2 ~b=c.
Given these asymptotes, the evolution matrix of system (17) is
diagonalizable00 with eigenvalues000 {l1’ and {l2’ and eigenvectors
respectively (l1 ,1=c)T and (c,l1 )T . Furthermore, it is not difficult
to see that
all00 those
l’s are asymptotically close to each other, that
0
000
is li ^li ^li ^li (i~1,2). We now define
so that Equation (8) can be rewritten as
dx. ~ðf(x. ,t){e Þdtz
ð16Þ
imply that the trajectory of any synchronized element of the network
xi and that of the noise-free system xnoise { free are also similar
[compare Figure 1(A) and Figure 1(E)].
Summing over i the equations followed by the xi and using
assumption (A1), we have
Ð1
ð15Þ
Finally, Equation (7) and the triangle inequality
Using (A4) then leads to
e~
ð14Þ
7
January 2010 | Volume 6 | Issue 1 | e1000637
How Synchronization Protects from Noise
0
these equations. Remark also that each pair (vij , wij ) is in fact
uncoupled with respect to other pairs. This allows us to use (19) to
obtain that, after transients of rate k,
00
(corollary 1 of [21]) and the approximations li ^li ^li , this
yields
8
<
:
(y21 )ƒs2 l1 ,
after transients of rate l1
2 2
c s
(y22 )ƒ
,
l2
Vi, j, i=j,
after transients of rate l2 :
X
8
1
c
>
>
>
< x1 ^ l1 y1 { l2 y2
1
r~
:
((w1 {w2 )2 )ƒ
s2 c3
:
bk2
ð19Þ
>
>
>
>
: dwij
n(n{1)s2
:
k
Since the (vi , wi )T are oscillators with stable limit cycles, it can be
shown that the trajectories of the vi are bounded by a common
constant M. Thus (A2) is verified with Hbd ~2cM.
Acknowledgments
The authors would like to thank the anonymous reviewers for their helpful
comments. JJS is grateful to Uri Alon for stimulating discussions on possible
relevance of the results to cell biology.
This publication reflects only the authors’ views. The European
Community is not liable for any use that may be made of the information
contained therein.
(c{(v2i zvi vj zv2j ){k)vij zcwij ) dt
pffiffiffi
z 2sdW
1
b
~
{ vij { wij dt:
c
c
~
n(n{1)s2
:
k
2cv 0
Hv ~
:
0 0
General case. Consider now an all-to-all network with
identical couplings as in Equation (2). Construct as above the
following n(n{1) auxiliary systems indexed by (i,j) [ ½1 . . . n2 ,
where the vi are considered as external inputs
8
>
dvij
>
>
>
<
ƒ
For large n, we have r=n2 *s2 =k, which converges to 0 when
k?? [see Figure 3(A)].
Assumption (A1) is also verified because an all-to-all network
with identical couplings is symmetric, therefore balanced. As for
(A2), observe that Hw ~0 and
Since (v1 {v2 , w1 {w2 )T is a particular trajectory of system
(17) as we remarked earlier, one finally obtains that, after
transients of rate k,
s2
k
(vi {vj )
2
Thus, (A4) is verified with
Thus, after transients of rate l1 ,
((v1 {v2 )2 )ƒ
!
ivj
1
1
>
>
>
: x2 ^{ 2 y1 z l y2 :
cl1
1
s2 c2
(x22 )ƒ 2
l1 l2
s2
:
k
Summing over the i, j yields
These bounds can be translated back in terms of the xi by
inverting (18)
s2
(x21 )ƒ
l1
((vi {vj )2 )ƒ
Author Contributions
Performed the experiments: NT. Contributed reagents/materials/analysis
tools: NT JJS QCP. Wrote the paper: NT JJS QCP.
Remark that, similarly to the case of two oscillators,
(vij , wij )T i,j ~ (vi {vj , wi {wj )T i,j is a particular solution of
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